scholarly journals Ando–Hiai-type inequalities for operator means and operator perspectives

2019 ◽  
Vol 31 (01) ◽  
pp. 2050007
Author(s):  
Fumio Hiai ◽  
Yuki Seo ◽  
Shuhei Wada
Keyword(s):  
Positive Operators ◽  
Extension Problem ◽  
Operator Means ◽  
Trotter Formula

We improve the existing Ando–Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie–Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

scholarly journals The logarithmic mean of two convex functionals

2020 ◽  
Vol 18 (1) ◽  
pp. 1667-1684
Author(s):  
Mustapha Raïssouli ◽  
Shigeru Furuichi
Keyword(s):  
Positive Operators ◽  
Logarithmic Mean ◽  
Convex Functionals ◽  
Operator Means ◽  
Theoretical Results

Abstract The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator versions of the functional theoretical results obtained here are immediately deduced without referring to the theory of operator means.

scholarly journals On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

2004 ◽  
Vol 11 (3) ◽  
pp. 479-487
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Plane ◽  
Identity Transformation ◽  
Extension Problem ◽  
Measure Extension ◽  
Nonmeasurable Set

Abstract The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑2 which are 𝑇2-negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇2 denotes the group of all translations of 𝐑2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇2, where 𝑔 is an arbitrary rotation of 𝐑2 distinct from the identity transformation and all central symmetries of 𝐑2.

On the isometric extension problem

2013 ◽  
Vol 36 (3) ◽  
pp. 321-330
Author(s):  
Ruidong Wang
Keyword(s):  
Extension Problem ◽  
Isometric Extension

Convexity Inequalities for Positive Operators

Positivity ◽  
2006 ◽  
Vol 11 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Markus Haase
Keyword(s):  
Positive Operators

On the End Extension Problem For Δ0-PA(S)

1989 ◽  
Vol 35 (5) ◽  
pp. 391-397
Author(s):  
Henryk Kotlarski
Keyword(s):  
Extension Problem

MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES

2015 ◽  
Vol 93 (3) ◽  
pp. 473-485 ◽  
Author(s):  
JIAN-ZE LI
Keyword(s):  
Banach Spaces ◽  
Equivalent Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extension Problem ◽  
Isometric Extension ◽  
Strictly Convex ◽  
Necessary And Sufficient ◽  
Strictly Convex Banach Spaces ◽  
Equivalent Conditions

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

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